Math Problem Statement

Show that the sequence \langlea_{n}\rangle defined as a_{n} = (((2n)!)/((n!) ^ 2)) ^ (1/n) converges to 4.

Solution

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Math Problem Analysis

Mathematical Concepts

Sequences and Series
Asymptotic Analysis
Combinatorics
Factorial Growth

Formulas

a_n = \left( \frac{(2n)!}{(n!)^2} \right)^{1/n}
Stirling's approximation: n! \sim \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n

Theorems

Stirling's Approximation
Limit of Sequences

Suitable Grade Level

College Level (Advanced Calculus or Combinatorics)